where \(M_*\)  is the running of the effective Planck mass. Such additional friction term also changes the damping rate of the GWs during their propagation. This leads to a GW luminosity distance \(d_L^{\rm gw}\)  which related to the standard luminosity distance of electromagnetic luminosity distance \(d_L^{\rm em}\)  as [1]}, [2]}
\(d_L^{\rm gw}(z) &=& d_L^{\rm em}(z) \exp \Big \lbrace \frac{1}{2} \int _0^{z} \frac{dz^{\prime }}{1+z^{\prime }} \frac{(\ln {\cal G}_0)^{\prime }}{\cal H}\Big \rbrace  \nonumber \\&=& d_L^{\rm em}(z) \exp \Big \lbrace  \frac{1}{2} \int d (\ln {\cal G}_0) \Big \rbrace .\) 
